Search for: All records

A homology class [Formula: see text] of a complex flag variety [Formula: see text] is called a line degree if the moduli space [Formula: see text] of 0-pointed stable maps to X of degree d is also a flag variety [Formula: see text]. We prove a quantum equals classical formula stating that any n-pointed (equivariant, [Formula: see text]-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety [Formula: see text]. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags [Formula: see text]. Our formulas make it straightforward to compute the big quantum [Formula: see text]-theory ring [Formula: see text] modulo the ideal [Formula: see text] generated by degrees d larger than line degrees. 

Abstract We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $$\rho $$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $$\rho =1$$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $$K$$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.